Question

Prove that the sum of the angles of a triangle is 180º. If the three angles of a triangle are in
the ratio 1:2:3 , find the angles.​

Answer 1

Step-by-step explanation:

1x+2x+3x=180°

6x=180°

x=30°

three angles are 30°,60°,90°.......

आगे आपकी मर्जी

Answer 2

Here, firstly we'll prove that the sum of the angles of a triangle is 180º & later we'll find out the 3 angles if they're in ratio 1 : 2 : 3, respectively.

I) Let us consider PQR is a triangle with angles ∠1, ∠2, ∠3.

Draw a line XY, which passes through line P, so that it will be parallel to QR.

Now, Consider lines XY & QR, where you'll observer PQ as a transversal.

Hence,

∠2 = ∠4 ⠀⠀⠀⠀[Alternate interior angles]

But from the figure, you can clearly specify that PR is also as a another transversal.

Hence,

∠3 = ∠5⠀⠀⠀⠀[Alternate interior angles]

Observe the line XY as well,

∠1 + ∠4 + ∠5 = 180°⠀⠀⠀⠀[linear pair of angles]

As we know that, ∠2 = ∠4 & ∠3 = ∠5.

Substituting the values,

∠1 + ∠2 + ∠3 = 180°

Hence Proved .

II) We're given with the ratio of angles, that is, 1 : 2 : 3, & we've to find out the values of angles.

Let us assume that the angles be x, 2x, 3x respectively !

By Angle sum property of a triangle,

$\sf \longrightarrow \: \angle1 + \angle2 + \angle3 ={180}^{ \circ}$

$\sf \longrightarrow \: x + 2x + 3x ={180}^{ \circ}$

$\sf \longrightarrow \: 6x ={180}^{ \circ}$

$\sf \longrightarrow \: x = \dfrac{180}{6}$

$\sf \therefore \large \: x = 30$

So, the required angles are :

• x = 30°

• 2x = 2(30)° = 60°

• 3x = 3(30)° = 90°